Inflated Cauchy Filters - A Way to Construct the Completion of a General Uniform Space

TL;DR

This paper introduces the concept of inflated Cauchy filters as a novel method for constructing the completion of a general uniform space without relying on pseudometrics, addressing a gap in classical topology.

Contribution

It proposes inflated Cauchy filters as a new approach to complete general uniform spaces, expanding the tools beyond traditional pseudometric-based methods.

Findings

Inflated Cauchy filters effectively construct uniform space completions.

The method aligns with Bourbaki's minimal Cauchy filters.

Provides a new perspective on uniform space completion without pseudometrics.

Abstract

Treatises about General Topology that emphasize the notion of uniformity and uniform space find, of course, no difficulty in defining the notion of a complete uniform space and in constructing the completion of a metric space, via its Cauchy sequences. In contrast, constructing the completion of a general uniform space, especially without recourse to pseudometrics, presents itself as somewhat awkward. In this note the notion of an inflated Cauchy filter is proposed as a way to accomplish that. As the author learned later, all that was actually expounded in Bourbaki, Topologie Generale, 1966 edition (where the filters are referred to as minimal Cauchy rather than inflated Cauchy), hence this note was withdrawn by the author.